Group #36470611 — subgroups

Subgroups — #36470611

Order 22784.

374 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 374 conjugacy classes of subgroups — too many to draw. Only its 131 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#15667952	C₂	1	1	yes	maximal, characteristic
2	C₂×#19151427	C₂	1	2	yes	maximal, complemented
3	#26020290	C₂	1	1	yes	maximal, characteristic
4	#26018345	C₂	1	2	yes	maximal, complemented
5	C₈₉×#1810	C₂	1	1	yes	maximal, characteristic
6	C₂×#15667952	—	2	4	no	—
7	C₂×#15667952	C₄	1	4	yes	—
8	C₂²×#13083549	C₂²	1	1	yes	characteristic
9	C₂²×#13083549	C₂²	1	1	yes	characteristic
10	C₂×#15667948	C₂²	1	2	yes	—
11	#19151562	—	2	1	no	—
12	#19151427	—	2	4	no	—
13	#19151363	—	2	1	no	—
14	C₂³×C₈×C₈₉	C₂²	1	1	yes	characteristic
15	C₂×#108×C₈₉	C₂²	1	2	yes	—
16	#407×C₈₉	—	2	1	no	—
17	#341×C₈₉	—	2	1	no	—
18	C₂×#13083558	—	2	2	no	—
19	#15667952	—	2	8	no	—
20	C₂×#13083549	—	2	2	no	—
21	C₂×#13083549	C₂×C₄	1	2	yes	—
22	C₂×#13083549	C₂×C₄	1	2	yes	—
23	C₂×#13083549	D₄	1	4	yes	—
24	#15667948	—	2	2	no	—
25	#15667948	—	2	2	no	—
26	C₂²⋊#10623494	—	4	2	no	—
27	C₂³×C₄×C₈₉	D₄	1	1	yes	characteristic
28	C₂×M₁₆×C₈₉	—	2	2	no	—
29	C₂²×C₈×C₈₉	—	2	2	no	—
30	C₂²×C₈×C₈₉	—	2	2	no	—
31	C₂²×C₈×C₈₉	C₂×C₄	1	2	yes	—
32	C₂²×C₈×C₈₉	D₄	1	1	yes	characteristic
33	C₂²×C₈×C₈₉	Q₈	1	2	yes	—
34	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
35	#108×C₈₉	—	2	2	no	—
36	#108×C₈₉	—	2	2	no	—
37	C₂²⋊C₈×C₈₉	—	4	2	no	—
38	#13083558	—	4	2	no	—
39	#13083558	—	4	2	no	—
40	C₂×#10623494	—	4	2	no	—
41	#13083549	—	2	2	no	—
42	#13083549	C₂²⋊C₄	1	4	yes	—
43	C₂⁴×C₈₉	Q₁₆	1	1	yes	characteristic
44	C₂²×C₄×C₈₉	—	2	2	no	—
45	C₂²×C₄×C₈₉	—	2	2	no	—
46	C₂²×C₄×C₈₉	C₄⋊C₄	1	2	yes	—
47	C₂²×C₄×C₈₉	D₈	1	2	yes	—
48	C₂²×C₄×C₈₉	Q₁₆	1	1	yes	characteristic
49	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
50	M₁₆×C₈₉	—	4	2	no	—
51	M₁₆×C₈₉	—	4	2	no	—
52	C₂×C₈×C₈₉	—	2	2	no	—
53	C₂×C₈×C₈₉	—	2	2	no	—
54	C₂×C₈×C₈₉	—	2	2	no	—
55	C₂×C₈×C₈₉	—	2	4	no	—
56	C₂×C₈×C₈₉	—	4	2	no	—
57	C₂×C₈×C₈₉	C₄⋊C₄	1	2	yes	—
58	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
59	C₂×C₈×C₈₉	C₂×D₄	1	2	yes	—
60	C₂×C₈×C₈₉	C₂×Q₈	1	1	yes	characteristic
61	C₂×C₈×C₈₉	#41	1	2	yes	—
62	C₂³×C₈₉	—	2	2	no	—
63	C₂³×C₈₉	—	4	2	no	—
64	C₂³×C₈₉	#107	1	2	yes	—
65	C₂³×C₈₉	C₂×Q₁₆	1	1	yes	characteristic
66	C₂×C₄×C₈₉	—	2	2	no	—
67	C₂×C₄×C₈₉	—	2	2	no	—
68	C₂×C₄×C₈₉	—	2	2	no	—
69	C₂×C₄×C₈₉	—	2	4	no	—
70	C₂×C₄×C₈₉	#107	1	2	yes	—
71	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
72	C₂×C₄×C₈₉	#121	1	2	yes	—
73	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
74	C₂×C₄×C₈₉	#135	1	2	yes	—
75	C₈×C₈₉	—	2	2	no	—
76	C₈×C₈₉	—	8	2	no	—
77	C₈×C₈₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
78	C₈×C₈₉	#117	1	1	yes	characteristic
79	C₈×C₈₉	C₂²⋊Q₈	1	2	yes	—
80	#10623494	—	8	2	no	—
81	C₂²×C₈₉	—	2	2	no	—
82	C₂²×C₈₉	—	2	2	no	—
83	C₂²×C₈₉	—	2	4	no	—
84	C₂²×C₈₉	—	4	4	no	—
85	C₂²×C₈₉	C₂×#107	1	1	yes	characteristic
86	C₂²×C₈₉	#467	1	2	yes	—
87	C₄×C₈₉	—	2	2	no	—
88	C₄×C₈₉	#380	1	1	yes	characteristic
89	C₄×C₈₉	#427	1	1	yes	characteristic
90	C₄×C₈₉	#466	1	2	yes	—
91	#12838	—	89	1	no	maximal, Sylow
92	C₂×C₈₉	—	2	2	no	—
93	C₂×C₈₉	—	4	2	no	—
94	C₂×C₈₉	#1808	1	1	yes	characteristic
95	C₂×C₈₉	#2161	1	2	yes	—
96	C₂²×M₃₂	—	89	1	no	—
97	C₂×#368	—	89	2	no	—
98	#1810	—	89	1	no	—
99	#1810	D₈₉	1	1	yes	characteristic
100	#1381	—	89	2	no	—
101	C₈₉	#12838	1	1	yes	characteristic, complemented, Sylow
102	C₂³×C₈	C₂×D₈₉	1	1	yes	characteristic
103	C₂×M₃₂	—	89	4	no	—
104	C₂×M₃₂	—	178	4	no	—
105	C₂²×C₁₆	—	89	1	no	—
106	C₂²×C₁₆	—	89	1	no	—
107	C₂×#108	—	89	2	no	—
108	C₂×#108	C₂×D₈₉	1	2	yes	—
109	#407	—	2	1	no	—
110	#407	—	178	1	no	—
111	#368	—	178	4	no	—
112	#341	—	2	1	no	—
113	#341	—	178	1	no	—
114	C₂³×C₄	C₂²⋊D₈₉	1	1	yes	characteristic
115	C₂×M₁₆	—	2	2	no	—
116	C₂×M₁₆	—	178	2	no	—
117	C₂²×C₈	—	2	2	no	—
118	C₂²×C₈	—	2	2	no	—
119	C₂²×C₈	Dic₁₇₈	1	2	yes	—
120	C₂²×C₈	C₄×D₈₉	1	2	yes	—
121	C₂²×C₈	C₂²⋊D₈₉	1	1	yes	characteristic
122	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
123	M₃₂	—	178	8	no	—
124	C₂×C₁₆	—	89	2	no	—
125	C₂×C₁₆	—	89	4	no	—
126	C₂×C₁₆	—	89	2	no	—
127	C₂×C₁₆	—	178	2	no	—
128	#108	—	2	2	no	—
129	#108	—	2	2	no	—
130	#108	—	178	2	no	—
131	#108	—	178	2	no	—
132	C₂²⋊C₈	—	4	2	no	—
133	C₂²⋊C₈	—	356	2	no	—
134	C₂⁴	#13083564	1	1	yes	characteristic
135	C₂²×C₄	—	2	2	no	—
136	C₂²×C₄	—	2	2	no	—
137	C₂²×C₄	#13083553	1	2	yes	—
138	C₂²×C₄	#13083559	1	2	yes	—
139	C₂²×C₄	#13083564	1	1	yes	characteristic
140	C₂²×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
141	M₁₆	—	4	2	no	—
142	M₁₆	—	4	2	no	—
143	M₁₆	—	356	2	no	—
144	M₁₆	—	356	2	no	—
145	C₂×C₈	—	2	2	no	—
146	C₂×C₈	—	2	2	no	—
147	C₂×C₈	—	2	4	no	—
148	C₂×C₈	—	2	2	no	—
149	C₂×C₈	—	4	2	no	—
150	C₂×C₈	—	356	2	no	—
151	C₂×C₈	#13083553	1	2	yes	—
152	C₂×C₈	C₂×C₄×D₈₉	1	1	yes	characteristic
153	C₂×C₈	D₄×D₈₉	1	2	yes	—
154	C₂×C₈	C₂×Dic₁₇₈	1	1	yes	characteristic
155	C₂×C₈	#13083573	1	2	yes	—
156	C₁₆	—	89	4	no	—
157	C₁₆	—	178	2	no	—
158	C₂³	—	2	2	no	—
159	C₂³	—	4	2	no	—
160	C₂³	#15667945	1	2	yes	—
161	C₂³	C₂×#13083564	1	1	yes	characteristic
162	C₂×C₄	—	2	2	no	—
163	C₂×C₄	—	2	2	no	—
164	C₂×C₄	—	2	2	no	—
165	C₂×C₄	—	2	4	no	—
166	C₂×C₄	#15667945	1	2	yes	—
167	C₂×C₄	C₂×#13083553	1	1	yes	characteristic
168	C₂×C₄	#15667984	1	2	yes	—
169	C₂×C₄	C₂×#13083559	1	1	yes	characteristic
170	C₂×C₄	#15668032	1	2	yes	—
171	C₈	—	2	2	no	—
172	C₈	—	8	2	no	—
173	C₈	—	712	2	no	—
174	C₈	C₂²⋊C₄×D₈₉	1	1	yes	characteristic
175	C₈	#15667968	1	1	yes	characteristic
176	C₈	#15667989	1	2	yes	—
177	C₂²	—	2	2	no	—
178	C₂²	—	2	4	no	—
179	C₂²	—	2	2	no	—
180	C₂²	—	4	4	no	—
181	C₂²	C₂×#15667945	1	1	yes	characteristic
182	C₂²	#19151837	1	2	yes	—
183	C₄	—	2	2	no	—
184	C₄	#19151456	1	1	yes	characteristic
185	C₄	#19151634	1	1	yes	characteristic
186	C₄	#19151832	1	2	yes	—
187	C₂	—	2	2	no	—
188	C₂	—	4	2	no	—
189	C₂	#26020279	1	1	yes	characteristic
190	C₂	#26022046	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×#15667952

C₂

yes

maximal, characteristic

C₂×#19151427

C₂

yes

maximal, complemented

#26020290

C₂

yes

maximal, characteristic

#26018345

C₂

yes

maximal, complemented

C₈₉×#1810

C₂

yes

maximal, characteristic

C₂×#15667952

—

C₂×#15667952

C₄

yes

—

C₂²×#13083549

C₂²

yes

characteristic

C₂²×#13083549

C₂²

yes

characteristic

C₂×#15667948

C₂²

yes

—

#19151562

—

#19151427

—

#19151363

—

C₂³×C₈×C₈₉

C₂²

yes

characteristic

C₂×#108×C₈₉

C₂²

yes

—

#407×C₈₉

—

#341×C₈₉

—

—

—

—

C₂×C₄

yes

—

C₂×#13083549

C₂×C₄

yes

—

C₂×#13083549

D₄

yes

—

#15667948

—

#15667948

—

C₂²⋊#10623494

—

C₂³×C₄×C₈₉

D₄

yes

characteristic

C₂×M₁₆×C₈₉

—

C₂²×C₈×C₈₉

—

C₂²×C₈×C₈₉

—

C₂²×C₈×C₈₉

C₂×C₄

yes

—

C₂²×C₈×C₈₉

D₄

yes

characteristic

C₂²×C₈×C₈₉

Q₈

yes

—

C₂²×C₈×C₈₉

C₂³

yes

characteristic

#108×C₈₉

—

#108×C₈₉

—

C₂²⋊C₈×C₈₉

—

—

—

—

—

C₂²⋊C₄

yes

—

C₂⁴×C₈₉

Q₁₆

yes

characteristic

C₂²×C₄×C₈₉

—

C₂²×C₄×C₈₉

—

C₂²×C₄×C₈₉

C₄⋊C₄

yes

—

C₂²×C₄×C₈₉

D₈

yes

—

C₂²×C₄×C₈₉

Q₁₆

yes

characteristic

C₂²×C₄×C₈₉

C₂×D₄

yes

characteristic

M₁₆×C₈₉

—

M₁₆×C₈₉

—

C₂×C₈×C₈₉

—

C₂×C₈×C₈₉

—

C₂×C₈×C₈₉

—

C₂×C₈×C₈₉

—

C₂×C₈×C₈₉

—

C₂×C₈×C₈₉

C₄⋊C₄

yes

—

C₂×C₈×C₈₉

C₂²×C₄

yes

characteristic

C₂×C₈×C₈₉

C₂×D₄

yes

—

C₂×C₈×C₈₉

C₂×Q₈

yes

characteristic

C₂×C₈×C₈₉

#41

yes

—

C₂³×C₈₉

—

C₂³×C₈₉

—

C₂³×C₈₉

#107

yes

—

C₂³×C₈₉

C₂×Q₁₆

yes

characteristic

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

#107

yes

—

C₂×C₄×C₈₉

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₈₉

#121

yes

—

C₂×C₄×C₈₉

C₂×D₈

yes

characteristic

C₂×C₄×C₈₉

#135

yes

—

C₈×C₈₉

—

C₈×C₈₉

—

C₈×C₈₉

C₂×C₂²⋊C₄

yes

characteristic

C₈×C₈₉

#117

yes

characteristic

C₈×C₈₉

C₂²⋊Q₈

yes

—

#10623494

—

C₂²×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

C₂×#107

yes

characteristic

C₂²×C₈₉

#467

yes

—

C₄×C₈₉

—

C₄×C₈₉

yes

C₄×C₈₉

yes

C₄×C₈₉

yes

—

#12838

—

maximal, Sylow

C₂×C₈₉

—

C₂×C₈₉

—

C₂×C₈₉

#1808

yes

characteristic

C₂×C₈₉

#2161

yes

—

C₂²×M₃₂

—

C₂×#368

—

#1810

—

#1810

D₈₉

yes

characteristic

100

#1381

—

101

C₈₉

#12838

yes

characteristic, complemented, Sylow

102

C₂³×C₈

C₂×D₈₉

yes

characteristic

103

C₂×M₃₂

—

104

C₂×M₃₂

—

178

—

105

C₂²×C₁₆

—

106

C₂²×C₁₆

—

107

C₂×#108

—

108

C₂×#108

C₂×D₈₉

yes

—

109

#407

—

110

#407

—

178

—

111

#368

—

178

—

112

#341

—

113

#341

—

178

—

114

C₂³×C₄

C₂²⋊D₈₉

yes

characteristic

115

C₂×M₁₆

—

116

C₂×M₁₆

—

178

—

117

C₂²×C₈

—

118

C₂²×C₈

—

119

C₂²×C₈

Dic₁₇₈

yes

—

120

C₂²×C₈

C₄×D₈₉

yes

—

121

C₂²×C₈

C₂²⋊D₈₉

yes

characteristic

122

C₂²×C₈

C₂²×D₈₉

yes

characteristic

123

M₃₂

—

178

—

124

C₂×C₁₆

—

125

C₂×C₁₆

—

126

C₂×C₁₆

—

127

C₂×C₁₆

—

178

—

128

#108

—

129

#108

—

130

#108

—

178

—

131

#108

—

178

—

132

C₂²⋊C₈

—

133

C₂²⋊C₈

—

356

—

134

C₂⁴

#13083564

yes

characteristic

135

C₂²×C₄

—

136

C₂²×C₄

—

137

C₂²×C₄

#13083553

yes

—

138

C₂²×C₄

#13083559

yes

—

139

C₂²×C₄

#13083564

yes

characteristic

140

C₂²×C₄

C₂×C₂²⋊D₈₉

yes

characteristic

141

M₁₆

—

142

M₁₆

—

143

M₁₆

—

356

—

144

M₁₆

—

356

—

145

C₂×C₈

—

146

C₂×C₈

—

147

C₂×C₈

—

148

C₂×C₈

—

149

C₂×C₈

—

150

C₂×C₈

—

356

—

151

C₂×C₈

#13083553

yes

—

152

C₂×C₈

C₂×C₄×D₈₉

yes

characteristic

153

C₂×C₈

D₄×D₈₉

yes

—

154

C₂×C₈

C₂×Dic₁₇₈

yes

characteristic

155

C₂×C₈

#13083573

yes

—

156

C₁₆

—

157

C₁₆

—

178

—

158

C₂³

—

159

C₂³

—

160

C₂³

#15667945

yes

—

161

C₂³

C₂×#13083564

yes

characteristic

162

C₂×C₄

—

163

C₂×C₄

—

164

C₂×C₄

—

165

C₂×C₄

—

166

C₂×C₄

#15667945

yes

—

167

C₂×C₄

C₂×#13083553

yes

characteristic

168

C₂×C₄

#15667984

yes

—

169

C₂×C₄

C₂×#13083559

yes

characteristic

170

C₂×C₄

#15668032

yes

—

171

C₈

—

172

C₈

—

173

C₈

—

712

—

174

C₈

C₂²⋊C₄×D₈₉

yes

characteristic

175

C₈

#15667968

yes

characteristic

176

C₈

#15667989

yes

—

177

C₂²

—

178

C₂²

—

179

C₂²

—

180

C₂²

—

181

C₂²

C₂×#15667945

yes

characteristic

182

C₂²

#19151837

yes

—

183

C₄

—

184

C₄

#19151456

yes

characteristic

185

C₄

#19151634

yes

characteristic

186

C₄

#19151832

yes

—

187

C₂

—

188

C₂

—

189

C₂

#26020279

yes

characteristic

190

C₂

#26022046

yes

—

	subgroup	G/H
center	C₂×C₄	C₂×#13083559
commutator	C₂×C₈×C₈₉	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₈₉
Fitting	C₈₉×#1810	C₂
socle	C₂²×C₈₉	C₂×#107
prime radical	C₂⁴×C₈₉	Q₁₆
cosocle	C₂²×C₈×C₈₉	C₂³
hypercenter	#1810	D₈₉
hypocenter	C₈₉	#12838
LCS	G ▷ C₂×C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ #1810

subgroup

G/H

center

C₂×C₄

C₂×#13083559

commutator

C₂×C₈×C₈₉

C₂²×C₄

Frattini

C₂²×C₈

C₂²×D₈₉

Fitting

C₈₉×#1810

C₂

socle

C₂²×C₈₉

C₂×#107

prime radical

C₂⁴×C₈₉

Q₁₆

cosocle

C₂²×C₈×C₈₉

C₂³

D₈₉

C₈₉

G ▷ C₂×C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ #1810

N	H ≅ G/N	type
2. C₂×#19151427	C₂	semidirect
4. #26018345	C₂	semidirect
101. C₈₉	#12838	semidirect

H ≅ G/N

type

2. C₂×#19151427

C₂

semidirect

4. #26018345

C₂

semidirect

101. C₈₉

#12838

semidirect

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂²×#15667952	2
C₂×#19151427	2
#26020290	2
#26018345	2
C₈₉×#1810	2
#12838	89

subgroup

index

C₈₉×#1810

Subgroups — #36470611

Conjugacy classes of subgroups

Complements

Maximal subgroups